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I have been using NASA's Program to Optimize Simulated Trajectories (POST) for developing a 3 DoF trajectory optimization code for ascent launch vehicle flight to orbit. The dynamics are integrated in Earth-Centered Inertial (ECI) frame where the thrust acceleration vector and the drag acceleration vector are modelled in Body (B) frame and transformed to ECI frame while the gravity acceleration vector is given in ECI frame directly. My understanding is that the B frame and geographic (G) frame (commonly known as North-East-Down frame) are rotating reference frames in contrast to ECI.

Translational Equations of Motion

The transformation between B and G frames are clear however for going from G to ECI or vice versa there has been some unidentified hiccups particularly for the acceleration vector transformation from B frame (to go to ECI frame) and relative velocity vector transformation from ECI (to go to B frame)

With this background, my questions are:

  1. Conventional knowledge says that for a vertically launched vehicle, the relative flight path angle (in G frame) should be 90 degree during vertical flight i.e. relative velocity vector in G-frame should be non-nulled in only one component. However, it is not so based on the above preliminaries, what might be the cause for this?

  2. Is there a particular component missing in the transformation of the thrust acceleration vector from G frame to ECI frame, besides the transformation matrix itself?

Transformation matrix for ECI to G-frame

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However, it is not so based on the above preliminaries, what might be the cause for this?

The linked paper does give a reason: "These specialized options are required to simulate certain physical constraints that are not modeled in the equations of motion." The physical constraint in the case of a vertical launch is very simple: The launch vehicle must not collide with the launch tower.

Many launch vehicles do not launch strictly vertically. They instead immediately tilt back a bit so that the thrust pushes the vehicle's nose a bit away from the launch tower. A bit later, they straighten back up to local vertical so that the tail will also clear the launch tower.

The impact of these very first maneuvers on the predicted optimal trajectory from algorithms such as POST are essentially in the noise. The easiest thing to do from a POST-like perspective is to assume that the rocket launches vertically (vertically from an Earth-centered Earth-fixed perspective) for the first ten to twenty seconds of the launch.

Is there a particular component missing in the transformation of the thrust acceleration vector from G frame to ECI frame, besides the transformation matrix itself?

No. A real force such as thrust (as opposed to a fictitious force) is the same vector across all Newtonian reference frames. This of course assumes Newtonian mechanics is valid. This is a valid assumption for the low velocities compared to the speed of light that occur during launch.

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